A Type Theory of Sense: Witnessed Choice in Stratified Semantic Spaces

Abstract: We introduce TTS, a dependent type theory in which semantic composition is represented by horn filling and distinctions between possible completions are witnessed relative to explicit measurement regimes. TTS replaces globally canonical composition with regime-indexed indiscernibility and constructive apartness, allowing filler spaces to be classified as canonical when all completions are observationally connected and forked when two warranted completions are positively separated. Separation witnesses enter the calculus only through measurement contexts recording actual instrument outputs, yielding conservativity, provenance, and a no-fork-from-the-empty-record result. We prove that forks persist under refinement while canonicity may fail, and characterize exactly when an identification made by one regime can consistently coexist with a separation made by another. This framework supports a geometric account of Fregean sense as a choice of filler, reference as the boundary constraining that choice, and hyperintensional difference as measured apartness, while providing a falsifiable bridge to stratified representation spaces and branching behaviour in language-model generation.